Devil in the Details: Molding, Stairs, and Tile to Carpet

Been all over the internet and have not found the answer to the questions below. I am tiling my bathroom. (I'm lacking all the tiling terms, so please excuse).

1. When I take out the molding around the edges of the floor, will I lay tile flush with the wall, and replace the molding? The tile will be a little higher than the linoleum-- so the molding if place on top of the tile would be higher... which then messes up all the molding. I worry about the molding around the door frame being fractions of an inch off as well.

2. A large walk-in closet opens up below the double sink-- two steps down. The tiling will stop abruptly at these two steps. What do I do to edge them? How to tile the top portion of the floor, which then goes into two carpeted steps?

3. My bathroom has a closed off section with a door for toilet and bath. The bedroom is carpeted, and opens up into a currently linoleum covered double sink area. Going from carpet to tile... what makes the transition? What holds the carpet down?

follow up question

I have a similiar followup question that perhaps someone could answer.

I am going to tile a kitchen & hallway that has the cellar door off of it. We have replaced the stairs down to the basement with stained wood treads up to but not including the top landing which overhangs like a stair tread, but is really part of the hallway floor.

How do I handle the tile there? If I bring the tile under the cellar door to the edge of the overhang like the current carpet does, we will see the tile and subfloor edges. I would prefer not to use the profile edge listed below if I don't have to. Any suggestions?